The globally irreducible representations of symmetric groups

Let $K$ be an algebraic number field and $\mathcal{O}$ be the ring of integers of $K$. Let $G$ be a finite group and $M$ be a finitely generated torsion free $\mathcal{O} G$-module. We say that $M$ is a globally irreducible $\mathcal{O}\, G$-module if, for every maximal ideal $\mathfrak{p}$ of $\mathcal{O}$, the $k_\mathfrak{p}\, G$-module $M\otimes _{\, \mathcal{O}} k_\mathfrak{p}$ is irreducible, where $k_\mathfrak{p}$ stands for the residue field $\mathcal{O}/\mathfrak{p}$.

Answering a question of Pham Huu Tiep, we prove that the symmetric group $\Sigma _n$ does not have non-trivial globally irreducible modules. More precisely we establish that if $M$ is a globally irreducible $\mathcal{O}\, \Sigma _n$-module, then $M$ is an $\mathcal{O}$-module of rank $1$ with the trivial or sign action of $\Sigma _n$.